The conditional distributions are very similar to what we have derived above, watch for HW question.
This can be done with least-squares methods or in a Bayesian framework.
We still need
krige function in gstat contains a function for kriging ; however, again this requires a known variogram.krige function in gstat contains a function for kriging ; however, again this requires a known variogram.## [using ordinary kriging]
## variofit: covariance model used is exponential
## variofit: weights used: npairs
## variofit: minimisation function used: optim
## Warning in variofit(emp.vario, cov.model = "exponential"): initial values
## not provided - running the default search
## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "0.84" "0" "0.17" "0.5"
## status "est" "est" "est" "fix"
## loss value: 6873.17015373233
## variofit: model parameters estimated by WLS (weighted least squares):
## covariance model is: exponential
## parameter estimates:
## tausq sigmasq phi
## 0.1797 0.8546 0.0000
## Practical Range with cor=0.05 for asymptotic range: 0.0001159668
##
## variofit: minimised weighted sum of squares = 6633.819
## variofit: covariance model used is exponential
## variofit: weights used: npairs
## variofit: minimisation function used: optim
## Warning in variofit(emp.vario2, cov.model = "exponential"): initial values
## not provided - running the default search
## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "1.14" "1.09" "0" "0.5"
## status "est" "est" "est" "fix"
## loss value: 2384.34387917209
## variofit: model parameters estimated by WLS (weighted least squares):
## covariance model is: exponential
## parameter estimates:
## tausq sigmasq phi
## 0.0189 1.0468 0.7845
## Practical Range with cor=0.05 for asymptotic range: 2.350179
##
## variofit: minimised weighted sum of squares = 1035.009
## Warning: Removed 35 rows containing non-finite values (stat_ydensity).
## Warning: Removed 43 rows containing non-finite values (stat_ydensity).
Distributional beliefs are iteratively updated in the presence of new data
A prior distribution, \(p(\boldsymbol{\theta}|\boldsymbol{\lambda})\) is the belief about the parameters(\(\boldsymbol{\theta}\)) before collecting data.
A set of hyperparameters, \(\boldsymbol{\lambda}\) can be used to define the prior distribution.
Write out and specify a sampling model for this regression problem.
Sketch and/or define a prior distribution for your parameters in the sampling model.
There are three levels of this (hierarchical) model
Bannerjee, Geland, and Carlin state, "Bayesian inferential paradigm offers potentially attractive advantages over the classical, frequentist statistical approach through
-The model for a Gaussian process can be written as \[Y(\boldsymbol{s}) = \mu(\boldsymbol{s}) + w(\boldsymbol{s}) + \epsilon(\boldsymbol{s}),\] where \(\mu(\boldsymbol{s}) = x(\boldsymbol{s})^t\boldsymbol{\beta}\) is the mean structure. Then the residual can be partitioned into two pieces: a spatial component \(w(\boldsymbol{s})\) and a non-spatial component \(\epsilon(\boldsymbol{s}).\)
We assume \(w(\boldsymbol{s})\) are realizations from a Gaussian Process (GP) with mean zero that models residual spatial structure.
Then \(\epsilon(\boldsymbol{s})\) are uncorrelated error terms.
Q: how do \(w(\boldsymbol{s})\) + \(\epsilon(\boldsymbol{s})\) relate to the partial sill, range, and nugget?
Next, prior distributions are necessary for